Uniform oscillation inequality for the sharply truncated Hilbert transform

Let $J_H(y)=\boldsymbol{1}_{[-1,1]}(y)y^{-1}$ be the Hilbert transform kernel with a sharp cutoff, and let $\operatorname{Osc}_n(J_H;f)$ denote the corresponding oscillation operator. Uniform oscillation conjecture. For every $n\geq 1$, every $1<p<\infty$, and every $f\in L^p$, one has

$$\|\operatorname{Osc}_n(J_H;f)\|_p\lesssim\|f\|_p.$$

Moreover, the implied constant can be chosen independently of $n$. This conjecture asks for uniform $L^p$ control of the oscillation operators associated with the sharply truncated Hilbert transform; the supplied text gives no evidence that it has been resolved.